Given:
#r(theta) = (theta)sin(theta) - cos^3(theta) + tan(theta), theta = pi/3#
Tangents with polar coordinates gives us the equation:
#dy/dx = ((dr(theta))/(d theta)sin(theta) + r(theta)cos(theta))/((dr(theta))/(d theta)cos(theta) - r(theta)sin(theta))#
#(dr(theta))/(d theta) = sin(θ)+θ cos(θ)+sec^2(θ)+3 sin(θ) cos^2(θ)#
The slope, m, is:
#m = ((dr(pi/3))/(d theta)sin(pi/3) + r(pi/3)cos(pi/3))/((dr(pi/3))/(d theta)cos(pi/3) - r(pi/3)sin(pi/3))#
#(dr(pi/3))/(d theta) = 4+(7 sqrt(3))/8+π/6#
#r(pi/3) = -1/8+sqrt(3)+πsqrt(3)/(6)#
#sin(pi/3) = sqrt(3)/2#
#cos(pi/3) = 1/2#
Substituting the above into the equation for m:
#m = ((4+(7 sqrt(3))/8+π/6)sqrt(3)/2 + (-1/8+sqrt(3)+πsqrt(3)/(6))(1/2))/((4+(7 sqrt(3))/8+π/6)(1/2) - (-1/8+sqrt(3)+πsqrt(3)/(6))sqrt(3)/2)#
Evaluation by WolframAlpha
#m ~~ 7.7#
